Problem: Let $g(x)=-4x^3+10x^2-x$. $g'(x)=$
Explanation: According to the sum rule, the derivative of $-4x^3+10x^2-x$ is the sum of the derivatives of $-4x^3$, $10x^2$, and $-x$. The derivatives of these terms can be found using the power rule : $\dfrac{d}{dx}(x^n)=n\cdot x^{n-1}$ For example, this is the derivative of the first term: $\begin{aligned}\dfrac{d}{dx}(-4x^3)&=-4\dfrac{d}{dx}(x^3)&&\gray{\text{Constant multiple rule}}\\\\ &=-4\cdot (3x^2)&&\gray{\text{Power rule}}\\ \\ &=-12x^2\end{aligned}$ Here is the complete differentiation process: $\begin{aligned} &\phantom{=}g'(x) \\\\ &=\dfrac{d}{dx}(-4x^3+10x^2-x) \\\\ &=-4\dfrac{d}{dx}(x^3)+10\dfrac{d}{dx}(x^2)-\dfrac{d}{dx}(x)&&\gray{\text{Basic differentiation rules}} \\\\ &=-4\cdot 3x^2+10\cdot2x-1x^0&&\gray{\text{The power rule}} \\\\ &=-12x^2+20x-1 \end{aligned}$ In conclusion, $g'(x)=-12x^2+20x-1$.